If there is no BIGGEST number, is there a SMALLEST?

[C1ay]

The epsilo-delta proof and notation of limit is perfectly fine. Traditional Calculus does NOT use nor define infinitesimals (nor does it use Infinity); Calculus just works with the Real Numbers. Other versions of Analysis do attempt to define infinitesimals. So, Calculus has little to contribute to this discussion, because Calculus (and limits) only deal with the finite.

You obviously did not study calculus from the same professor I did. I specifically remember dealing infinitesimal quantites in differential calculus and we quite frequently dealt with infinity as well. I recommend you find another instructor.

[Qfwfq]

All I am saying is that BOTH an infinitesimal (1/INF) and the Absolute Zero are solutions to the equation:

 

a + X = a , a is an integer

Not quite. The difference is very subtle. In any case infinitesimal isn't defined by 1/inf, it's a very different story.

 

If e is an infinitesimal term, a + e is not equal to a. The limit of it is a, in the same limit in which e is zero. No infinitesimal term is a solution of the equation a + x = a.

 

With that understood, we can go on to hypothesize that maybe, we do NOT need to define the Absolute Zero, as we do not need to define an Absolute Infinity.

 

(This a weaker response.

Actually, we need to leave the Absolute Infinity undefined for consistency).

If you really want to define Absolute Infinity because "Absolute" Zero (as you call it) is defined, then you could define Absolute Infinity as the reciprocal of Absloute Zero = 1/0.

 

I doubt you will change the discipline of mathematics in this way because there is really no need for the definition, there is no lack of consistency. The fact that infinity isn't an element of R doesn't even make R topologically incomplete, simply because a sequence having infinite limit isn't a Cauchy sequence.

[Guest loarevalo]

Not quite. The difference is very subtle. In any case infinitesimal isn't defined by 1/inf, it's a very different story.

 

If e is an infinitesimal term, a + e is not equal to a. The limit of it is a, in the same limit in which e is zero. No infinitesimal term is a solution of the equation a + x = a.

 

If you really want to define Absolute Infinity because "Absolute" Zero (as you call it) is defined, then you could define Absolute Infinity as the reciprocal of Absloute Zero = 1/0.

 

I doubt you will change the discipline of mathematics in this way because there is really no need for the definition, there is no lack of consistency. The fact that infinity isn't an element of R doesn't even make R topologically incomplete, simply because a sequence having infinite limit isn't a Cauchy sequence.

 

Some of you seem confused about what I am arguing for, and what I rely on as "established" facts in mathematics. Because these "facts" may not be so widely known as I thought, I include a reference.

 

*FACTS:*

 

* There are many sizes of infinity. There is a marked difference between the potential infinity of Arithmetic and Calculus, and the actual infinities dealt in Set Theory:

http://en.wikipedia.org/wiki/Cardinal_number

http://en.wikipedia.org/wiki/Aleph_number

 

* The Absolute Infinity - Biggest number - set containing all sets does NOT exist. The very definition of it is inconsistent: To quote from the source, "there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy. Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set."

http://en.wikipedia.org/wiki/Absolute_infinite

 

* Infinity plus one is Infinity. That fact I think dwells on the obvious. Aleph_a +1 = Aleph_a (Example: Aleph_null +1 = Aleph_null):

http://mathworld.wolfram.com/Aleph-0.html

 

As we see, Infinity is very complex. Should we not also allow that complexity to infinitesimals?

 

I argue for the same that the FACTS establish for infinity; only, that instead of infinity, I apply them to the infinitesimals, like a mirror. All I am arguing for is symmetry.

 

- Since there are many sizes of infinity,

therefore there are many sizes (degrees) of infinitesimals

(Leibnez and Euler also argued this).

 

- Since INF + 1 = INF,

therefore infinitesimal + 1 = 1

 

- Since Absolute Infinity does not exist

therefore Absolute Zero does not exist.

 

(I consider the terms "zero" and "infinity" equally vague. I also use the term "zero" and "infinitesimal" interchangeably. If you say "zero," I know you probably mean "Absolute zero")

 

One last FACT:

 

* Calculus does not use actual infinitesimals (it does not use them!). Instead, Calculus uses limits. To quote from source, "[non-standard analysis] provides a rigorous justification of purely formal calculations using infinitesimals to derive facts about derivatives, integrals, and series. Such formal calculations with infinitesimals were widely used before alternative and rigorously justified methods, without infinitesimals were introduced in the 19th century." That alternative and rigorous method that needs neither an actual infinity nor infinitesimals is the "limit."

http://en.wikipedia.org/wiki/Non-standard_calculus

[Guest loarevalo]

I suppose I agree on cataloging this claim "strange" and thus being sent here. Even more strange I guess is to dare give "facts" that in a way may support the claim - besides just philosophical babling mumbo jumbo.

 

All I am asking is:

Doesn't it make sense?

 

I guess the answer is "no" - the claim is way past common sense.

[alxian]

besides 0 then you are talking numbers to quantify small things, the smallest being strings if you are so inclined. however being that strings are theory as atoms were theory forever will we be finding smaller things to quantify.

 

regardless of the physical though and into the realm of the purely theoretical you could always find something to quantify that uses ever larger negative notations. its a matter of what you are quantifying and more importantly what you are comparing it to. how many strings in a given galaxy for instance, thats 1: (a very big number). is it useful? not right now but once quantum and subquantum teleportation becomes practical you'll need to know the relative positions and states of all quantum coordinates within and between given teleportation points. thats quite a large amount of data, mostly repeated (prolly on the fly compressed for more effiecient crunching)... otherwise and on a smaller scale revising the human genome may require large amounts of data meaning the smaller elements (the amino acids) will have small numbers attached to them vs full chains of tissues and organs and whole creatures. (think batman or sharkman but no suits. hmmm eating but to fuel a 160kg flying metabolism or eating tuna to fuel a 200kg shark man metabolism.. can't decide.)

[Guest loarevalo]

Thank you for that. It was amusing. I needed it. But I can't see your point no matter how hard I try. Thank you though - for that bit of humor.

[C1ay]

Infinitesimal

 

In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. A number x ≠ 0 is an infinitesimal iff every sum |x| + ... + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number. Infinitesimals, obviously, are not real numbers, so "operations" on them aren't familiar.

[Qfwfq]

Trust Wikipedia. That's not the definition in standard calculus, in which there's no such thing as what they describe and there is no definition of an infinitesimal number.

 

An infinitesimal term is one having 0 as some limit. Since lim(x-->0) x = 0, x is infinitesimal in 0. Trivial. So also is 1/x infinitesimal at infinity.

 

They mention hyperreal numbers, which I have heard of, which contemplate infinitely large and infinitesimal numbers as actually being elements of the field. Perhaps Loarevalo would be interested in those, personally I don't find it very useful to extend R this way. Everything in calculus can be done as rigorously as one pleases with limits, or made handier with the use of symbols such as dx which is a linear form, and not a number.

[C1ay]

Trust Wikipedia. That's not the definition in standard calculus, in which there's no such thing as what they describe and there is no definition of an infinitesimal number.

So you disagree with "Infinitesimals, obviously, are not real numbers, so "operations" on them aren't familiar." My point was only to show you can't do operations on them. If x is an infinitesimal and the limit as x->0 is 0 then is x/x=1 or undefined? The limit is 0 but x is not. It was always my understanding that you just can't do operations with infinitesimals.

 

Hyperreal numbers, numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0, are covered in non-standard analysis. Perhaps there Loarevalo will find some interests, but even there x<>0.

[Guest loarevalo]

So you disagree with "Infinitesimals, obviously, are not real numbers, so "operations" on them aren't familiar." My point was only to show you can't do operations on them.

 

Perhaps there Loarevalo will find some interests, but even there x<>0.

 

I don't get it. What Wikipedia was saying in regard to infinitesimals was their operations were not "familiar" by which is meant that we - shallow minded people - aren't familiar with those operations. You can do operations with infinitesimals - they're just strange, like operations with infinities.

 

The definition that Wikipedia gives of infinitesimals is fine. I just don't think it may be for the best to assert that for an infinitesimal A:

 

A + 1 > 1

 

when one accepts:

 

INF + 1 = INF

 

If INF +1 > INF, in all cases, then we wouldn't be talking about infinity would we?

So, If A+1 > 1, then A is still not a "true" infinitesimal, only a number that is less than all reals, or less than all hypperreals, or less than ...

 

I have read the actual logical constructions of systems (hypperreals, surreals, NSA) and frankly found them somewhat artificial - forced and unnatural. The construction of infinity through Set Theory is most natural (common sense appealing) construction I know of, and I wouldn't expect anything less from a construction of infinitesimals.

 

I still think there must be a direct symmetric relation between Infinity and the Infinitesimal. Does anyone at least agree on that?

[C1ay]

The definition that Wikipedia gives of infinitesimals is fine.

So you are saying that it is OK that an infinitesimal is a number that is greater in absolute value than zero yet smaller than any positive real number, thus an infinitesimal x ≠ 0?

Does x/x = 1 because x > 0 or is it undefined because x = 0?

[Dark Mind]

That last question threw me.

I'll go read up on more than just the previous five posts before I ask what you mean though :hihi:.

[Guest loarevalo]

So you are saying that it is OK that an infinitesimal is a number that is greater in absolute value than zero yet smaller than any positive real number, thus an infinitesimal x ≠ 0?

Does x/x = 1 because x > 0 or is it undefined because x = 0?

 

Yes. I agree in that an Infinitesimal is greater in absolute value than the Absolute Zero. As I said before, when you say "0" I know you probably mean the "Absolute Zero."

 

Yes. An infinitesimal x > 0, if by "0" you precisely mean only the Absolute Zero.

No. An infinitesimal x =0, If by "0" you mean the value x, for which n+x=n. That is, by "0" you are referring to the more general and vague form.

 

If that totally confused you, Let me say the same only now for Infinity.

 

For an actual infinity x, like the count of how many natural numbers:

 

x < ∞ , if by "∞" you precisely mean only the Absolute Infinity (the BIGGEST number)

x = ∞ , if by "∞" you mean the broad concept of "infinite," that is, you are referring to all forms of infinity. Obviously by, x +n = x, one can see that all forms of infinity and not only the Absolute Infinity, satisfy this equation.

 

I hope this wasn't too confusing :)

[nkt]

If you have an infitessimally small number, and you multiply it by infinity, it must be infinity, since only infinity * 0 = 0. A simple mathmatical form for this infintesimally small number would simply be 1/infinity. I have no idea what use it could be, however.

[Guest loarevalo]

If you have an infitessimally small number, and you multiply it by infinity, it must be infinity, since only infinity * 0 = 0. A simple mathmatical form for this infintesimally small number would simply be 1/infinity. I have no idea what use it could be, however.

 

Yes. Thank you for your openess.

 

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

 

Some wouldn't want to define infinitesimals directly from infinity, as you did: 1/INF. In my view, if you don't define infinitesimals by way of infinity, you're not really defining infinitesimals. As far as I know, mathematics has NOT yet defined TRUE infinitesimals.

 

Even though you may agree with

 

1 + x = 1 , x being an infinitesimal x =1/INF.

 

most mathematicians don't agree with that. The evidence: Mathematics still says that for any "infinitesimal" x: 1+x > 1. Yet, ironically, for an infinite X: X+1=1.

 

That's the issue.

[Dark Mind]

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

How???

[nkt]

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

It should always be zero.

The evidence: Mathematics still says that for any "infinitesimal" x: 1+x > 1. Yet, ironically, for an infinite X: X+1=1.

As X tends to infinity, X+1 also tends to infiinity.

 

No irony required.